Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for $\mathcal{O}$-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for $\mathcal{O}$-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for $\mathcal{O}$-blocks is a consequence of conjectures predicting bounds on the $\mathcal{O}$-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.


Introduction
Let p be a prime and let k =F p . Let (K , O, k) be a p-modular system, so O is a complete discrete valuation ring with residue field k. The results here hold in this general setting, but we have in mind for O the ring of Witt vectors over k as this will be used to state Donovan's conjecture in a uniform way. Donovan's conjecture, originally stated over an algebraically closed field, is as follows: The paper is structured as follows. In Sect. 2 we recall the definition of the strong O-Frobenius number and state some of the main results. In Sect. 3 we show that Donovan's conjecture for O-blocks is equivalent to two separate conjectures as in [12]. Section 4 contains the reduction to quasisimple groups. In Sect. 5 we briefly discuss the problem of bounding Cartan invariants.
Remark on choice of O in Donovan's conjecture: Note that since O/J (O) is algebraically closed we ensure that K contains all p -roots of unity. In general O would have to contain a primitive |P| th root of unity in order for K to be a splitting field for a block with defect group P, but this condition is not always necessary to demonstrate Donovan's conjecture. We therefore have two canonical choices for O in the statement of Donovan's conjecture: the ring of Witt vectors forF p and the same with a primitive |P| th root of unity attached. The results of this paper hold over either choice (see Remark 4.7 for the latter case), but in light of the results of [9] the former seems the best setting for Donovan's conjecture.

Strong O-Frobenius and O-Morita-Frobenius numbers
The strong O-Frobenius number was introduced in [4], but we recall the definition and some of its main properties here. We also define the O-Morita-Frobenius number. These numbers may be defined for any choice of O as in the introduction, although this requires some care when it comes to defining the character idempotents.
Let A be an O-algebra finitely generated as an O-module. Write k A for A ⊗ O k and K A for A ⊗ O K . Let G be a finite group and B a block of OG. Denote by e B ∈ OG the block idempotent corresponding to B and e k B ∈ kG the block idempotent corresponding to k B. Write Irr(G) for the set of irreducible characters of G and Irr(B) for the subset of Irr(G) of irreducible characters χ such that χ(e B ) = 0. For each χ ∈ Irr(G) we denote by e χ ∈ K G the character idempotent corresponding to χ, where K denotes the algebraic closure of K . Note that K B = χ ∈Irr(B) K Ge χ . If X and Y are finitely generated Ralgebras for R ∈ {K , O, k, K }, we write X ∼ Mor Y if the categories of finitely generated X and Y -modules are (Morita) equivalent as R-linear categories.
Denote by π a generator of the maximal ideal of O. Let σ ∈ Gal(K /Q p ) be such that σ (π) = π and σ induces a non-trivial automorphismσ on O/π O = k.
Define A (σ ) to be the O-algebra with the same underlying ring structure as A but with a new action of the scalars given by λ.a = λ σ −1 a, for all λ ∈ O and a ∈ A. We may similarly define (k A) (σ ) . We note that, through the identification of elements, A and A (σ ) are isomorphic as rings but not necessarily as O-algebras. In the case thatσ is the Frobenius automorphism given by x → x q , where q is a power of p, it is sometimes convenient to write − (q) forσ . If B is a block of OG, for some finite group G, then we can also define B (q) to be B (σ ) , where σ is some lift of − (q) . We define B (q) in an alternative way below. In particular we show that it is independent of the choice of σ .
For a general σ we have OG (σ ) ∼ = OG and we can realise this isomorphism via: If B is a block of OG, then we identify B (σ ) with its image under the above isomorphism. We can do analogous identifications with kG and its blocks. Let q = p z for some z ∈ Z. By an abuse of notation we use − (q) to also denote the field automorphism of the universal cyclotomic extension of Q defined by ω p ω p → ω p ω q p , for all p th -power roots of unity ω p and p th roots of unity ω p . If χ ∈ Irr(G), then we define χ (q) ∈ Irr(G) to be given by χ (q) (g) = χ(g) (q) for all g ∈ G. If B is a block of OG with χ ∈ Irr(B), then we define B (q) to be the block of OG with χ (q) ∈ Irr(B (q) ). We have

Definition 2.1 Let
A be a finitely generated k-algebra.
Let B a block of OG, for some finite group G.
Note that the definition of strong O-Frobenius number given above is not exactly the same as that given in [4,Definition 3.8] but the two are shown to be equivalent in [4,Proposition 3.5].
A consequence of the following is that bounding the strong O-Frobenius numbers in terms of the size of the defect group is equivalent to bounding the O-Morita-Frobenius numbers in terms of the size of the defect group.

Proposition 2.2 Let G and H be finite groups, and let B and C be blocks of OG and O H respectively. Let D be a defect group for B.
by π a generator of the maximal ideal of O. Let us fix an element σ of Gal(K /Q p ) such that σ (π) = π and σ induces a positive power of the Frobenius automorphism on O/π O. If O is the ring of Witt vectors over k, then π = p and any power of the Frobenius automorphism of k can explicitly be lifted to O. We denote the automorphism of k that σ induces byσ . The ultimate aim of the section is to prove an analogue over O for Kessar's results in [12] which hold over k.
Defining "− σ ", resp."− σ ", to be the elements fixed under σ , resp.σ , the field k σ is finite by definition and we claim that (K σ , O σ , k σ ) is again a p-modular system. It is clear that K σ is complete and that O σ is integrally closed in K σ . Moreover O σ /πO σ ⊆ k σ . We just need to check that this inclusion is an equality. To see this, note that every non-zero element of k σ is a (|k σ | − 1) th root of unity, and those lift to O by Hensel's lemma. That is, the groups of (|k σ | − 1) th roots of unity of O and k are in bijection (via reduction mod π), and sinceσ acts trivially on the latter, σ must act trivially on the former. Hence (|k σ | − 1) th roots of unity in O lie in O σ and reduce to the non-zero elements of k σ , so the claim is shown.
If we set C n+1 := C n − π n · X · C n , then we have The same congruence mod π n is satisfied by assumption. Thus we can rewrite this as We can find such an X once we show that the map is surjective. Surjectivity of this map is equivalent to the statement that the polynomial equation x − x q − z = 0 has a solution in k for any z ∈ k. Since k is algebraically closed, such a solution always exists. Therefore, by induction, there exist

Theorem 3.3 Let be an
Proof As a set (σ ) is equal to , and hence we may view as a σ -sesquilinear map from into itself. Now fix an isomorphism of O-lattices : denote the σ -sesquilinear map given by entry-wise application of σ . Then the map is O-linear (being the composition of a σ -sesquilinear and a σ −1 -sesquilinear map), and may therefore be viewed as an element of GL n (O) (as all maps involved in its construction are bijections). Now Lemma 3.2 implies that there is an A ∈ GL n (O) such that

The above equation implies that
Let e 1 , . . . , e n denote the standard basis of O n , and set The σ -sesquilinearity of implies that The fact that is multiplicative (by virtue of being an algebra isomorphism between and σ ) implies that Since the λ i are linearly independent it follows that m i, j;l = σ (m i, j;l ) for all i, j, l ∈ {1, . . . , n}, i.e. m i, j;l ∈ O 0 . This implies that the O 0 -lattice spanned by λ 1 , . . . , λ n is an O 0 -algebra, which completes the proof.
In the following we have in mind the case K 0 = K σ .

Proposition 3.4
Given a finite extension K 0 /Q p , and a natural number n, there are only finitely many isomorphism classes of semi-simple K 0 -algebras A of dimension n.
Proof The Artin-Wedderburn theorem implies that it suffices to prove that there are only finitely many division algebras A of a given dimension n over K 0 . As a Z (A)-algebra, a skew-field A is determined by its Hasse invariant (see [18, § 14]), which can take only finitely many possible values once we fix dim Z (A) (A). Hence we are reduced to showing that there are only finitely many possibilities for the field Z (A), that is, that there are only finitely many field extensions of K 0 of degree at most n. But it is well known that the number of extensions of Q p of a fixed degree is finite (see [13,Théorème 2], which even gives an explicit formula). Clearly the same holds for extensions of K 0 , as K 0 is of finite degree over Q p . This completes the proof.
In what follows we denote by length R (M) ∈ Z ≥0 ∪ {∞}, for a commutative local ring R and R-module M, the length of M as an R-module. We will also allow more flexibility for the choice of K 0 . We will often ask that K 0 /Q p be an extension of discretely valued fields, which means that it should be a (not necessarily finite) field extension such that the usual discrete (exponential) valuation ν p : Q p −→ Z ⊂ Q satisfying ν p ( p) = 1 extends to a discrete valuation K 0 −→ Q, also denoted by ν p . It is well known that the valuation on Q p can be extended (even uniquely) to any algebraic extension of finite degree. But K /Q p is an extension of discretely valued fields as well, after appropriate rescaling of the valuation. To be explicit, we let ν p : K −→ Q denote the unique discrete valuation on K such that ν p ( p) = 1. If we equip K with this valuation, K /Q p becomes an extension of discretely valued fields. Proposition 3.5 Let K 0 /Q p be an extension of discretely valued fields, let O 0 be the associated discrete valuation ring, and let A be a split semisimple K 0 -algebra. We have Denote by Tr i : A −→ K 0 the trace function on the i th matrix algebra summand of A. Fix elements u 1 , . . . , u n ∈ K × 0 and define Proof All maximal orders in A are conjugate. Moreover, any conjugate of is self-dual with respect to the same bilinear form T , that is, (a a −1 ) = a a −1 for any a ∈ A × . This is because the trace functions Tr i used in the definition of T are invariant under conjugation. Hence we may assume without loss of generality that Using the matrix units as an explicit basis of we can compute Moreover, T induces a non-degenerate pairing (with values in K 0 /O 0 , the quotient of the underlying additive group of K 0 by the underlying additive group of O 0 ) between the O 0modules / and / = / . It follows that these two O 0 -modules have the same length, which must consequently be exactly half the length of / . The asserted formula for the length of / now follows immediately. Definition 3.6 (Defect of a symmetric order) Let K 0 /Q p be an extension of discretely valued fields and let O 0 be the associated discrete valuation ring.
1. Let A be a split semisimple K 0 -algebra. We have for certain u 1 , . . . , u n ∈ K × 0 such that = (see [16, Definition (III.1)] for an introduction to symmetrising forms from this point of view). We call 2. Assume now that A is an arbitrary semisimple K 0 -algebra and that, as in the previous point, ⊆ A is a symmetric O 0 -order. Then there is an algebraic extension E 0 /K 0 of finite degree such that E 0 ⊗ K 0 A is split. As the extension is of finite degree, the discrete valuation of K 0 extends uniquely to a discrete valuation on E 0 . If E 0 denotes the valuation ring of E 0 , then E 0 ⊗ O 0 is an E 0 -order in the split semisimple E 0 -algebra E 0 ⊗ K 0 A, and we define the defect of to be equal to the defect of E 0 ⊗ O 0 (which is defined as per the previous point).
Remark 3.7 1. Note that the defect of a symmetric order is well-defined (i.e. independent of the choice of T and the splitting field E 0 ). Independently of whether K 0 is a splitting field for A, a symmetrising form T defines an isomorphism of − -bimodules. Such an isomorphism is clearly unique up to an automorphism of viewed as a --bimodule, and such automorphisms are given by multiplication by an element of Z ( ) × . So if T is another symmetrising form for , then T (−, =) = T (z · −, =) for some z ∈ Z ( ) × . If K 0 is a splitting field for A, then for all i and all a, b ∈ A we have Tr i (zab) = z i Tr i (ab) for some z i ∈ O × 0 (using the notation of Definition 3.6). In particular, the u i attached to the forms T and T differ only by an element of O × 0 , that is, they have the same valuation. The above argument shows that the defect of a symmetric order in a split semisimple algebra is independent of the choice of a symmetrising form. The second part of Definition 3.6 defines the defect in the non-split case by passing to a splitting field. So assume that we have two different splitting fields E 0 and E 0 , both of finite degree over K 0 . We need to show that the defect of is independent of whether we use E 0 or E 0 as our splitting field in Definition 3.6. Fix an algebraic closureK 0 of K 0 . We can choose embeddings i : E 0 →K 0 and i : E 0 →K 0 . Then there is a bigger splitting field E 0 ⊂K 0 containing both i(E 0 ) and i(E 0 ). As the valuation ν p on K 0 extends uniquely to any finite algebraic extension, we have ν p (i(x)) = ν p (x) for all x ∈ E 0 (same for i and E 0 ). Hence we may replace, without loss of generality, E 0 by i(E 0 ) and E 0 by i (E 0 ) and just assume that E 0 and E 0 are contained (as discretely valued fields) in E 0 . The explicit symmetrising forms T and T we chose over E 0 and E 0 both extend linearly to symmetrising forms over E 0 . The invariants u i used in Definition 3.6 for T (respectively T ) are the same as for the E 0 -linear extension of T (respectively T ). That is, the defect of obtained using the splitting field E 0 (respectively E 0 ) is the same as the one obtained using the splitting field E 0 . As we have seen in the previous paragraph that the defect of an order in a split semi-simple algebra over a given field is well-defined, it follows that defect of defined using the splitting fields E 0 or E 0 is the same. 2. Let E 0 /K 0 be an extension of discretely valued fields, and let E 0 and O 0 denote the corresponding discrete valuation rings. If is an O 0 -order in a semisimple K 0 -algebra, then the defect of the O 0 -order is the same as the defect of the E 0 -order E 0 ⊗ O 0 (this is again easy to see, one just needs to construct a finite splitting extension of E 0 containing a finite splitting extension of K 0 ). 3. If e ∈ is an idempotent, and T : A × A −→ K 0 is a symmetrising form for , then T | e Ae×e Ae is a symmetrising form for e e. In particular, if e does not annihilate any non-zero element of Z (A), then and e e have the same defect (this can be seen by passing to a splitting field). It follows that the basic algebra of has the same defect as , that is, the defect is invariant under Morita equivalence. 4. Let = O 0 G, and assume without loss of generality that K 0 G is split. If χ 1 , . . . , χ n : K 0 G −→ K 0 are the (absolutely) irreducible characters of G, then χ i = Tr i (up to permutation of the indices). It is easy to see that O 0 G is self-dual with respect to the bilinear form T (a, b) = |G| −1 ·χ reg (a ·b), where χ reg denotes the character of the regular representation of G. We have χ reg = i χ i (1) · χ i , and therefore That is u i = |G| −1 · χ i (1). In particular, the defect of O 0 G is equal to ν p (|G|). 5. If = O 0 Gb is a block, then the above reasoning implies that the defect of in the sense of Definition 3.6 is equal to This equals the defect of O 0 Gb in the ordinary sense (that is, the p-valuation of the order of a defect group) since any block contains a character of height zero.
Proof If A is split then this follows immediately from Eq. (2). If A is not split, E 0 is a finite extension of K 0 which splits A and E 0 is the integral closure of O 0 in E 0 , then (the ramification index of the extension E 0 /K 0 ). The right hand side can be bounded using Eq. (2) as before, so Theorem 3.9 Fix d, n ∈ N. Up to isomorphism there are only finitely many symmetric Oorders satisfying all of the following: Proof Define K 0 = K σ and O 0 = O σ . By Theorem 3.3 any satisfying the conditions above has an O 0 -form 0 . By Proposition 3.4 there are only finitely many K 0 -algebras which can occur as the K 0 -span of 0 . Hence it suffices to show that any semisimple K 0 -algebra A 0 contains only finitely many isomorphism classes of symmetric O 0 -orders of defect d.
The algebra A 0 contains a maximal order 0 which is unique up to conjugation. By Proposition 3.8 the O 0 -length of the quotient 0 / 0 for a symmetric O 0 -order 0 ⊆ 0 of defect d is bounded by 1 2 · e · d · n, where e = length O 0 (O 0 / pO 0 ). Now we just need to realise that there are only finitely many isomorphism classes of O 0 -modules of length smaller than this bound (as the residue field of O 0 is finite), and for each of these (torsion) O 0 -modules the set of O 0 -homomorphisms from 0 onto the module is a finite set. Any 0 occurs as the kernel of such a homomorphism, which proves that there are only finitely many possibilities.

The O-Morita-Frobenius number of the block is bounded by m.
Proof Consider the basic algebra of such a block, note that this is also symmetric with the same defect. The bound on the Cartan numbers implies a bound on the dimension of the K -span of the basic algebra. Moreover, a Morita equivalence of blocks corresponds to an isomorphism of basic algebras. Let n denote the least common multiple of the integers between 1 and m, and let σ be a lift of the n th power of the Frobenius automorphism of k. Any basic algebra of a block satisfying the third condition will satisfy ∼ = (σ ) , since n is a multiple of the O-Morita-Frobenius number of . It follows that the collection of basic algebras of the blocks satisfying the three conditions satisfies the conditions of Theorem 3.9 (for the chosen σ ). Hence this collection contains only finitely many isomorphism classes of orders, which implies the assertion of the theorem.

Corollary 3.11 Let X be a collection of O-blocks of finite groups and let P be a finite p-group.
The following are equivalent:

Reductions for Donovan's conjecture
The general strategy for the reduction for Donovan's conjecture is the same as that in [5], where the reduction proceeds in two steps. First it is shown that it suffices to consider reduced pairs, and then it is shown that in order to demonstrate the conjecture for reduced pairs, we need only consider quasisimple groups. In [5] the first part of the reduction, to reduced pairs, could only be achieved over k since it relied on the results of [14]. However the analogue of the results of [14] has since been shown by the second author. The following comes from [7,Corollary 4.18]. Our definition of reduced pairs is as in [5]: Definition 4.2 Let G be a finite group and B a block of OG with defect group D. We call (G, B) a reduced pair if it satisfies the following: We now give the first part of the reduction, which is analogous to [5,Proposition 3.4] and based on [2]. We give a proof here for completeness.

Proposition 4.3 Let P be an abelian p-group for a prime p. In order to verify Donovan's conjecture for P, it suffices to verify that there are only a finite number of Morita equivalence classes of blocks B of OG with defect group D ∼ = P occurring in reduced pairs (G, B).
Proof Fix a finite abelian p-group P.
Consider pairs ([G : O p (Z (G))], |G|) with the lexicographic ordering, where G is a finite group. We first use two processes, labelled (a) and (b), which we apply alternately to O-blocks of finite groups with defect groups isomorphic to P. Both processes strictly reduce ([G : O p (Z (G))], |G|) when applied non-trivially, hence after repeated application must terminate.
Let G be a finite group and B be a block of OG with defect group D ∼ = P. corresponding to c J is isomorphic to a block of Xi∈J L i with a central p -group in the kernel.
Products of nilpotent blocks are nilpotent, so c J is nilpotent. Since O p (G) ≤ Z (G), by [19] c J is also a nilpotent block, of a nonsolvable normal subgroup covered by C, a contradiction. Hence for all j we have D ∩ L j Z (L j ), so (R4) holds for (H , C). Conjugation induces a permutation action on the components, hence a homomorphism ϕ : H → S t . Let g ∈ D and say L g i = L j for some i = j. Since D ∩ L i Z (L i ) and L i ∩ L j ⊆ Z (L i ) we have a contradiction and so D ≤ ker(ϕ). Now (R5) implies that ker(ϕ) = H , i.e., (R3) holds for (H , C), and (H , C) is reduced.
Finally, by Theorem 4.1 for a fixed Morita equivalence class for C, there are only finitely many possibilities for the Morita equivalence class of B, and we are done.
In the second part of the reduction, from reduced pairs to blocks of quasisimple groups, we first show that in order to bound the strong O-Frobenius number for reduced pairs it suffices to bound it for quasisimple groups. This is already given in [5]: The remainder of the proof of Theorem 1.2 now consists of observing that bounding the strong O-Frobenius numbers for reduced pairs implies a bound on the number of Morita equivalence classes amongst reduced pairs. In [5], this part of the reduction could only be achieved over k since it relied on the results of [12]. The results of the previous section remedy this.  (G, B) where B is a block of OG with abelian defect group D. We have assumed that the Cartan invariants of the blocks of quasisimple groups with abelian defect groups are bounded in terms of the defect. Then by [2,Theorem 3.2] the Cartan invariants of any block with abelian defect groups are bounded in terms of the defect, and so in particular this holds for blocks B for finite groups G such that (G, B) is reduced. Hence by Theorem 3.11 there are only finitely many Morita equivalence classes amongst blocks in reduced pairs with defect group isomorphic to P and by Proposition 4.3 we are done.  Proof This follows immediately from Corollary 4.5 and [6, 9.2], in which it is proved that the Cartan invariants of 2-blocks with abelian defect groups are bounded in terms of the defect. Therefore all the results of this section hold for O equal to the ring of Witt vectors of k adjoining a primitive |P| th root of unity, where we are considering blocks with defect group isomorphic to P. This is a very common and natural choice of ring to work over as it guarantees that e χ ∈ K B for all χ ∈ Irr(B).

Bounding Cartan invariants
We are left with the difficult problem of finding a bound on the Cartan invariants of blocks of quasisimple groups in terms of the defect group, so we gather together some (known) comments on the problem. In general, there are few p-groups for which a bound on the Cartan invariants is known but Donovan's conjecture is not known to hold. The generalised quaternion 2-groups are an exception, where Donovan's conjecture is still not known in the case where the block has two simple modules, but the Cartan matrix is known (see [8]). Following [2], for G a finite group and B a block of kG with defect group D, write LL(B) for the Loewy length of B, the smallest n such that rad n (B) = 0. Write e(B) = max{dim k (Ext 1 kG (V , W )) : V , W simple kG−modules}.